Question

Weekly expenditure in an office for three weeks is given as follows. Assuming that the salary in all the three weeks of different categories of staff did not vary, calculate the salary for each type of staff, using the matrix inversion method.

Week	Number of employees			Total weekly salary (in ₹)
	A	B	C
1st week	4	2	3	4900
2nd week	3	3	2	4500
3rd week	4	3	4	5800

Answer 1

Let ₹ x, ₹ y, ₹ z be the salary for each type of staff A, B and C.

4x + 2y + 3z = 4900

3x + 3y + 2z = 4500

4x + 3y + 4z = 5800

The given system can be written as

`[(4,2,3),(3,3,2),(4,3,4)][(x),(y),(z)] = [(4900),(4500),(5800)]`

AX = B

where A = `[(4,2,3),(3,3,2),(4,3,4)]`, X = `[(x),(y),(z)]` and B = `[(4900),(4500),(5800)]`

|A| = `|(4,2,3),(3,3,2),(4,3,4)|`

= 4(12 – 6) – 2(12 – 8) + 3(9 – 12)

= 4(6) – 2(4) + 3(-3)

= 24 – 8 – 9

= 7

[A_ij] = `[(6,-4,-3),(-|(2,3),(3,4)|,|(4,3),(4,4)|,-|(4,2),(4,3)|),(|(2,3),(3,2)|,-|(4,3),(3,2)|,|(4,2),(3,3)|)]`

= `[(6,-4,-3),(-(8-9),16-12,-(12-8)),((4-9),-(8-9),(12-6))] = [(6,-4,-3),(1,4,-4),(-5,1,6)]`

adj A = [A_ij]^T = `[(6,1,-5),(-4,4,1),(-3,-4,6)]`

`"A"^-1 = 1/|"A"|` (adj A)

`= 1/7[(6,1,-5),(-4,4,1),(-3,-4,6)]`

X = A^-1B

`[(x),(y),(z)] = 1/7[(6,1,-5),(-4,4,1),(-3,-4,6)] [(4900),(4500),(5800)]`

`= 100/7 [(6,1,-5),(-4,4,1),(-3,-4,6)] [(49),(45),(58)]`

`= 100/7 [(294 + 45 - 290),(-196 + 180 + 58),(-147 - 180 + 348)]`

`= 100/7[(49),(42),(21)]`

`[(x),(y),(z)] = [(700),(600),(300)]`

∴ Salary for each type of staff A, B and C are ₹ 700, ₹ 600 and ₹ 300.